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Expert-verified Found in: Page 1 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Show that any positive definite matrix A can be written as , where B is a positive definite matrix.

$A={B}^{2}$ , where B is a positive definite matrix.

See the step by step solution

## Step 1: Given Information:

$A={B}^{2}$

## Step 2: Determining is the B is positive definite Matrix:

Consider the quadratic form $\mathrm{q}\left(\stackrel{\to }{\mathrm{x}}\right)=\stackrel{\to }{\mathrm{x}}·\mathrm{A}\stackrel{\to }{\mathrm{x}}$, where A is a $\mathrm{n}×\mathrm{n}$ symmetric matrix. If $\mathrm{q}\left(\mathrm{x}\right)$ is positive for all nonzero $\stackrel{\to }{x}$ in ${\mathrm{ℝ}}^{n}$, we say A is positive definite. S is an orthogonal matrix that has the following properties:

${S}^{-1}AS=D$

When D is a diagonal matrix, all of the entries are positive. As a result, A can be written as:

$A=SD{S}^{-1}$

We can write D as a diagonal matrix with positive diagonal elements.

$D={D}_{1}^{2}$

Where is a diagonal matrix with positive diagonal entries, Equation can now be written as follows:

$A={\mathrm{SDS}}^{-1}$

$⇒A=S\left({D}_{1}^{2}\right){S}^{-1}\phantom{\rule{0ex}{0ex}}⇒A=S\left({D}_{1}·{D}_{1}\right){S}^{-1}\phantom{\rule{0ex}{0ex}}⇒A=\left(S{D}_{1}{S}^{-1}\right)·\left(S{D}_{1}{S}^{-1}\right)\phantom{\rule{0ex}{0ex}}⇒A={B}^{2}$

${\mathrm{SBS}}^{-1}={\mathrm{D}}_{1}$ is a diagonal matrix with positive diagonal elements, hence B is positive definite.

## Step 3: Determining the Result:

$A={B}^{2}$, where B is a positive definite matrix. ### Want to see more solutions like these? 